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So from my understanding for a finite difference approximation, you're supposed to expand the series "about" the point $x$, e.g.,

$$u(x+h) = u(x) + h \ u'(x) + \frac{1}{2}h^2u''(x)+\frac{1}{6}h^3u'''(x)+O(h^4)$$

So I'm confused as to the "about" part. How do you know how many terms to expand out to exactly? If I were thinking simply, I'd say there should be 5 terms, so that two would be on either side, then one in the middle for symmetry, but this clearly isn't the case, as seen above.

My question is how do you determine how many terms to use, and why?

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  • $\begingroup$ What numerical method are you doing this for? $\endgroup$ – naslundx Mar 2 '14 at 21:40
  • $\begingroup$ finite difference approximations $\endgroup$ – helposaurus Mar 2 '14 at 21:51
  • $\begingroup$ You include enough terms so that you can subtract the terms you need, and the term of lowest order which remains is used for your truncation error. If you include more details as to what you're trying to calculate here I can be more specific and helpful. $\endgroup$ – naslundx Mar 2 '14 at 21:53
  • $\begingroup$ I'm just going over the general theory, nothing in specific. It's just never explained in plain english, and is not obvious to me. Basically we are shown $D_+$, $D_-$ and $D_0$, which are the forward backward and centered approximations. Then from there they say, well you can expand out $u(x+h)$ and $u(x-h)$ like so, with this many Taylor terms, and that's where they lost me. $\endgroup$ – helposaurus Mar 2 '14 at 22:01

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